Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems
For numerical analysts, it provides a theoretical framework for achieving high accuracy in ill-conditioned problems using preconditioning.
This paper shows that if a preconditioner's inverse can be applied accurately, ill-conditioned linear systems can be solved accurately, and applies this to compute smallest eigenvalues of ill-conditioned matrices.
This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an ill-conditioned linear system $Ax=b$, we show that, if the inverse of the preconditioner $M^{-1}$ can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse-equivalent accuracy is introduced to describe higher accuracy that is achieved and an error analysis will be presented. As an application, we use the preconditioning approach to accurately compute a few smallest eigenvalues of certain ill-conditioned matrices. Numerical examples are presented to illustrate the error analysis and the performance of the methods.